Elementary arithmetic, with brief notices of its history... by Robert Potts.

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each other, can be expressed in lower terms by dividing the numerator
and denominator by any common divisor; and into their lowest terms
by dividing them by the greatest common measure or divisor.
12. PROP. To reduce two or mzore fractions having different denominators
to other equivalent frations, each having the same common denominator.
It will be always found most convenient in practice to employ the
least common denominator.
1. Let the denominators be prime to each other and the fractions
in their simplest form.
If there be two fractions:
Multiply the numerator and denominator of the first fraction by the
denominator of the second; and the numerator and denominator
of the second fraction by the denominator of the first.
But, if there be more than two:
Multiply the numerator and denominator of each fraction by the
product of the denominators of the rest.
2. If the denominators be not prime to each other:
79= 1 1 - -1 the equivalent continued fraction.
443 2+2+9+2+4
It will be obvious that the process above is the same as that of finding the
greatest common divisor of the terms of the fraction, and that the successive
quotients, 2, 2, 9, 2, 4, form the successive denominators of the continued fraction.
Conversely, the continued fraction + - may be reduced to a single
fraction by reversing the process by which the continued fraction was found, begining with the last fraction in the series.
The successive simple fractions found by taking one, two, &c., terms of a continued fraction, form a series of fractions successively approximating to the value
of the continued fraction.
Find the series of fractions each of which converges to the value of the continued
11111
fraction 1
2+2+9+2+44
One term - differs from 79 by 85 in excess.
2 443 886
2 b 9
Two terms 1t1= |e2 by 2- in defect.
Two terms 2215
1 1 1 19 4by 4
Three terms 1 1 9 by in excess.
2+ + 9 47 20821
1 1 1 r1 40 1
Four terms by 57in dfect.
2 2 + 9 + 2- 99 43857
1 1 1 1 1 179
And the whole five terms + 9 + I the total value of the continued
2~2~9+2+4 443'
fraction.
The third converging fraction -7 may be found by a very simple process. Since
19=9 x 2 + 1, and 47= 9 x 5 +2; it appears that 19, the numerator of the third
fraction, is equal to the product of 2, the numerator of the second fraction, and 9,
the third quotient, added to 1, the numerator of the first fraction: and 47, the denominator of the third fraction, is equal to the product of 5, the denominator of the
second fraction, and 9, the third quotient, added to 2, the denominator of the first
fraction. And in a similar way may be found the fourth, &c., converging fractions.